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The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.

The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.

The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.

In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

This paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.

A rigorous error analysis is done, and optimal L² error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.

Optimal L² error estimates and energy norm.

The goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.

The purpose of this study is to investigate the interaction between magnetotactic bacteria and Fe₃O₄–water nanofluid (NF) in a wavy enclosure in the presence of 2D natural convection flow.

Uniform magnetic field (MF), Brownian and thermophoresis effects are also contemplated. The dimensionless, time-dependent equations are governed by stream function, vorticity, energy, nanoparticle concentration and number of bacteria. Radial basis function-based finite difference method for the space derivatives and the second-order backward differentiation formula for the time derivatives are performed. Numerical outputs in view of isolines as well as average Nusselt number, average Sherwood number and flux density of microorganisms are presented.

Convective mass transfer rises if any of Lewis number, Peclet number, Rayleigh number, bioconvection Rayleigh number and Brownian motion parameter increases, and the flux density of microorganisms is an increasing function of Rayleigh number, bioconvection Rayleigh number, Peclet number, Brownian and thermophoresis parameters. The rise in buoyancy ratio parameter between 0.1 and 1 and the rise in Hartmann number between 0 and 50 reduce all outputs average Nusselt, average Sherwood numbers and flux density of microorganisms.

This study implies the importance of the presence of magnetotactic bacteria and magnetite nanoparticles inside a host fluid in view of heat transfer and fluid flow. The limitation is to check the efficiency on numerical aspect. Experimental observations would be more effective.

In practical point of view, in a heat transfer and fluid flow system involving magnetite nanoparticles, the inclusion of magnetotactic bacteria and MF effect provide control over fluid flow and heat transfer.

This is a scientific study. However, this idea may be extended to sustainable energy or biofuel studies, too. This means that a better world may create better social environment between people.

The presence of magnetotactic bacteria inside a Fe₃O₄–water NF under the effect of a MF is a good controller on fluid flow and heat transfer. Since the magnetotactic bacteria is fed by nanoparticles Fe₃O₄ which has strong magnetic property, varying nanoparticle concentration and Brownian and thermophoresis effects are first considered.

The study explores how the intention to export affects financing and non-financing variables for small and medium-sized enterprises (SMEs) in a low-income country (LIC). The objectives of this study are (1) to discern between regional and global exporting and (2) to evaluate its policymaking implications.

Primary survey data were collected from 330 Rwandan SMEs and were analysed using ordered logistic models as an application of the expectation-maximisation iterating algorithm, which was tested for robustness using a sampling model variation.

The results show that alternative sources of finance are the predominant choice to finance the intention to export within and outside Africa. As the scope of export intentions broadened from regional to global, there was a shift in preferences from less formal to more formal lending technologies, moving from methods like factoring to lines of credit. Moreover, reliance on bank officers became more significant, with increasing marginal effects. Finally, the study determined that government financing schemes were not relevant for SMEs pursuing either regional or global exporting.

Whilst alternative sources of finance predominate the export intentions of Rwandan SMEs, establishing a robust banking relationship becomes crucial for global exporting. Despite this implication, the intention to export should prompt more transparent communication regarding government financial support programmes. There is an opportunity for increased usage of relationship lending to customise support for SMEs involved in exporting, benefiting both the private and public sectors.

This study accentuates how export distance alters SME financing priorities. The results also contribute to understanding how the value of relationship lending changes when less familiar markets (i.e. global exporting) are the objective. Moreover, the study offers a new perspective on how institutional voids affect entrepreneurial financing decisions in LICs.

The purpose of this research was to investigate at the influence of technology acceptance model (TAM) characteristics including perceived ease of use, perceived compatibility, and perceived security on the perceived utility and actual use of mobile banking among Yemeni academics. It also investigated the function of perceived utility as a moderator in the link between TAM variables and mobile banking adoption. The sample size was 251 respondents who worked at universities in Yemen and were chosen using stratified random selection. According to the findings, perceived compatibility and perceived security had a substantial positive influence on perceived usefulness and real mobile banking use; however perceived simplicity of use had no effect on mobile banking usage. According to the results, perceived usefulness slightly mediated the association between perceived ease of use and security and mobile banking use, whereas it completely mediated the relationship between perceived compatibility and mobile banking usage. Overall, the research proved the significance of TAM elements in academics’ use of mobile banking in Yemen, offering a credible empirical framework for studying mobile banking usage in this setting.

The public healthcare system faces challenges and limitations regarding the supply and delivery of healthcare services. The private healthcare system is expensive and only affordable for some. Due to the increasing population, developing countries face a greater degree of demand–supply mismatch. The existing healthcare services in developing countries need to be more sustainable due to high out-of-pocket expenditures and low-income levels. The research design used in this chapter is a case study approach based on qualitative data. The study focuses on two objectives: (1) to have a detailed understanding of the nature of healthcare cooperatives in Spain and their contribution to meeting healthcare requirements; and (2) to make suggestions and recommendations for an improved and sustainable healthcare cooperative for developing countries. Inspired by the healthcare cooperative model of Spain and keeping in mind the ground reality of the lack of healthcare facilities and services accessible and affordable in developing countries, the authors have developed a conceptual framework with the foundation of an insurance cooperative. The Spain Model is sustainable for developing countries as it serves the interest of all income brackets, not leaving the low-income population behind. Concepts like cooperative worker insurance embedded in the model can ensure that the beneficiaries receive timely and good quality health services at an affordable price. It empowers individuals by allowing them to make small amounts of investments as premiums to secure a hopeful future for a healthy life.

This paper aims to assess the feasibility of additively manufactured wind tunnel models. The additively manufactured model was used to validate a computational framework allowing the evaluation of the performance of five wing models.

An optimized fighter wing was additively manufactured and tested in a low-speed wind tunnel to obtain the aerodynamic coefficients and deflections at different speeds and angles of attack. The flexible wing model with optimized curvilinear spars and ribs was used to validate a finite element framework that was used to study the aeroelastic performance of five wing models. As a computationally efficient optimization method, homogenization-based topology optimization was used to generate four different lattice internal structures for the wing in this study. The efficiency of the spline-based optimization used for the spar-rib model and the lattice-based optimization used for the other four wings were compared.

The aerodynamic loads and displacements obtained experimentally and computationally were in good agreement, proving that additive manufacture can be used to create complex accurate models. The study also shows the efficiency of the homogenization-based topology optimization framework in generating designs with superior stiffness.

To the best of the authors’ knowledge, this is the first time a wing model with curvilinear spars and ribs was additively manufactured as a single piece and tested in a wind tunnel. This research also demonstrates the efficiency of homogenization-based topology optimization in generating enhanced models of different complexity.

This study aims to test negotiation outcomes when bilinguals negotiate in a foreign rather than their native language. Decision research on the foreign language effect indicates that bilingual individuals may be less susceptible to framing bias when using a foreign language because they make less emotional and biased choices. With increasing international business activity, there is a pressing need to examine the effect of language on bilingual negotiators.

The authors tested the hypotheses using a two (task frame: gain vs loss) × 2 (language: foreign vs native) factorial design recruiting 246 Korean–English bilinguals. A negotiation simulation with three issues was used, and participants exchanged offers with a preprogrammed computer they believed to be a real counterpart.

There was no significant interaction effect between framing and language on the offers made, but the framing effect was mitigated and nonsignificant for negotiators who used their foreign language. The interaction between framing and language conditions significantly affected negotiators’ positive emotions and satisfaction with the negotiation.

The uniqueness of this paper is related to its effort to investigate the effect of negotiation language on a negotiator’s decision-making. Considering globalization and the increasing prevalence of international negotiations, this paper has implications for researchers and practitioners.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.

The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.

The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.

Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.